4 DERIVATIVES OF FUNCTIONS OF ONE VARIABLE 157 Show that the sequence (/„) converges uniformly in I towards a continuous function which has no derivative at any point of I (use (a)). 2. Let /be a continuous mapping of an open interval I <=• R into a Banach space E, which has at every point / e I both a derivative on the left £(0 and a derivative on the right /*'«. (a) Let U be a nonempty open subset of E, A the set of points t e I such that /JO) £ U. For any a > 0, let Ba be the subset of I consisting of points t such that there is at least a point s e I for which t - oc ^ s < t and (/(/) ~f(s))/(t - s) e U; show that Ba is open and that A n (JB« is denumerable (use Problem 3 of Section 3.9). Conclude from that result that the set of points t e A such that/;(f) $ U is at most de- numerable. (b) Deduce from (a) that the set of points t e I such that/J(/) ^f'A(i) is at most de- numerable. (Observe first that /(I) is a denumerable union of compact metric spaces, and by considering the closed vector subspace of E generated by /(I), reduce the problem to the case in which the topology of E has a denumerable basis (Un) of open sets; then remark that for every pair of distinct points a, b of E there is a pair of sets UP5 Uq such that a e Up, b e U, and Vp n U, - 0.) 3. (a) Let / be defined in R2 by the conditions for * = &>&)* (0,0), /(0) = 0. Show that for any x E R2 and any y e R2, the limit lim (f(x +ty) — f(x))/t = g(x,y) t-+0, t^O exists but that y~*g(Q,y) is not linear (hence / is not differentiate at the point 0). (b) Let /be defined in R2 by the conditions /« = j%I for * = (&,&)* (0,0), Show that the limit g(x, y) exists for every x and y and y -*g(x, y) is linear for every x e R2, but that/is not differentiable at the point 0. (Consider the points (f i, f2) such that £2 = f ?.) 4. (a) Let / be a continuous mapping of an open subset A of a Banach space E into a Banach space F. We say that at x0 e A the function/is quasi-differentiable if there exists a linear mapping u of E into F, having the following property: for any continuous mapping g of I = [0,1 ] into A such that #(0) = x0 and that the derivative #'(0) of g at 0 (with respect to I) exists, then t -*f(g(t)) has at the point t = 0 a derivative (with respect to I) equal to u(g'(Q)). The linear mapping u is then called a quasi-derivative of/at x0. Show that if / is quasi-differentiable at x0, its quasi-derivative is unique. Extend property (8.2.1) to quasi-differentiable mappings, (b) Show that if / is quasi-differentiable at *o, its quasi-derivative u is a continuous linear mapping of E into F. (Suppose, as one may, that XQ = 0,/U0) = 0. Use contra- diction: if u is not bounded in the ball B(0; 1), there exists a sequence (#„) of vectors in E such that ||a,,II— 1, and a sequence (tn) of numbers >0, such that lim /n = 0 and n-*oo that \\tnlf(tnaM\ = <*n tends to +00; one can suppose that the sequences (tn) and (V^n) are decreasing and tend to 0. Define a continuous mapping of [0,1] intoE such that #(0) = 0, that ^'(O) exists and is equal to 0, and that g(\]